The X-Files certainly influenced Supernatural's style and storytelling, and quite a few future Supernatural notables appeared on the earlier show. To be more accurate though, The X-Files influenced just about every show dealing with the supernatural and paranormal to be made since, especially on network TV. Small-screen stories being told through a combination of serialized main plots and "monster of the week" standalone stories have remained a thing ever since Mulder and Scully pioneered the format, although it's gotten less common in the age of 13-episode or less seasons.
To its credit, Supernatural has never been shy about acknowledging its debt to The X-Files, directly referencing the show on multiple occasions, and doing subtle nods to it that only fans would recognize even more often. While Sam and Dean Winchester may not be romantic interests like Mulder and Scully - much to the chagrin of part of the fanbase, as the show has made fun of - their mix of banter and teamwork is certainly partly inspired by them.
Beyond just influences though, Supernatural has also employed quite a few actors that had appeared on The X-Files previously, over a hundred in fact. For the sake of brevity though, the round-up of crossover stars below only includes people who played an important, multi-episode role on Supernatural.
Mark Pellgrino
Mark Pellegrino is best known for playing Lucifer on Supernatural, as well as his most common vessel, Nick. On The X-Files, Pellegrino played Derwood Spinks, in season 7's "Hungry," an ex-con who makes the mistake of blackmailing a monster and gets eaten alive for it. Pellegrino also played the mysterious Jacob on Lost.
Mark Sheppard
Long before Mark Sheppard was charming Supernatural fans as devilish villain - and later antihero - Crowley, the future King of Hell was appearing on a season 1 X-Files episode called "Fire." He played lead bad guy Cecil Lively, a murderous psychopath with pyrokinesis, which is really not a good thing for anyone who crosses him.
Jim Beaver
On Supernatural, Jim Beaver plays beloved supporting character Bobby Singer, a surrogate father to Sam and Dean. Beaver appeared in the season 6 X-Files episode "Field Trip," a popular fan pick for best of that season. When Mulder disappears investigating a case, Scully works with the local coroner to find him, played by Beaver.
Mitch Pileggi
Mitch Pileggi's presence is the most direct onscreen connection between The X-Files and Supernatural, as he played major roles on both programs. Pileggi of course became a household name as Walter Skinner, Mulder and Scully's FBI boss, and later played Samuel Campbell, Sam and Dean's resurrected grandfather.
Steven Williams
Steven Williams is a very prolific character actor, but Mr. X, a mysterious government informant that leaks information to Mulder, is probably his most famous role. He also played charismatic bounty hunter Creighton Duke in Jason Goes to Hell. On Supernatural, Williams played Rufus Turner, a close friend and hunting partner to Bobby.
Rachel Miner
The briefest of Supernatural and X-Files actor crossovers on this list goes to Rachel Miner, who played most fans' favorite incarnation of the demon Meg. Meg once shared a kiss with Castiel, and to say fans were excited would be an understatement. Miner played a brief uncredited appearance as an FBI analyst in X-Files season 3's "Pusher."
Nicki Aycox
Speaking of Meg, Nicki Aycox played an earlier incarnation of Meg on Supernatural. On The X-Files, she appeared in the season 7 episode "Rush," playing a character with the rather exotic dancer-ish name of Chastity Raines. She's one of several high school students who discover a mysterious cave that grants them super speed.
Amanda Tapping
On Supernatural, Amanda Tapping plays angel big shot Naomi, who has had a very complex relationship with Castiel, to put it mildly. Still, she's shown to care deeply about protecting Heaven's souls. On The X-Files, Tapping played Carina Sayles in season 3's "Avatar," who has a one night stand with Skinner, but ends up dead afterward.
Emily Perkins
Emily Perkins first got noticed as young Beverly Marsh in the 1990 IT miniseries, then later played one of the leads in the cult werewolf film Ginger Snaps. On Supernatural, Perkins played Becky Rosen, an in-universe super-fan of Sam and Dean. On The X-Files, she played Roberta Dyer, a half-angel/half-human, in season 5's "All Souls."
Fredric Lehne
Another consummate character actor, Fredric Lehne boasts well over 100 credits on his resume. On Supernatural, he played Azazel, the yellow-eyed demon who killed Sam and Dean's mom. Over on The X-Files, he appeared in the episodes "Travelers" and "The Unnatural," as a young version of early X-Files office investigator Arthur Dales.
Ty Olsson
On Supernatural, Ty Olsson played Benny Lafitte, a vampire that Dean bonds with while trapped in purgatory, with the two coming to rely on each other for survival. Olsson played the small role of an orderly in the season 5 X-Files episode "Kitsunegari," which was actually a sequel to the aforementioned "Pusher."
Keith Szarabajka
Keith Szarabajka plays Donatello Redfield, a put-upon prophet of the lord, on Supernatural. He previously played cult leader Anthony Tipet on the season 8 X-Files episode "Via Negativa," one of the best of the Doggett era. Fans would be forgiven if they didn't retroactively recognize him though, as he looks almost completely different now.
Benito Martinez
Benito Martinez is another familiar TV face, having played notable roles on shows like House of Cards, Sons of Anarchy, The Shield, and more. On Supernatural, Martinez played Edgar, a high-ranking lieutenant to Leviathan leader Dick Roman. Martinez played an orderly in the season 6 X-Files episode "The Beginning."
Chad Lindberg
Chad Lindberg recurred as mullet-wearing tech expert Ash on Supernatural, based out of the Roadhouse bar operated by Sam and Dean's friends Jo and Ellen Harvelle. In X-Files season 5 episode "Schizogeny," Lindberg played Bobby Rich, a troubled teen that at first appears to be responsible for a few deaths.
A.J. Buckley
On Supernatural, AJ Buckley has recurred as Ed Zeddemore, a self-styled paranormal investigator and leader of the "Ghostfacers" group. His name is obviously a reference to Winston Zeddemore from Ghostbusters. Buckley previously played a small part dubbed "Dude," in X-Files season 3's "War of the Coprophages."