From what I know, the X planes have always been Air Force, Navy or NASA funded experimental aircraft or rockets.
Back in the day, it was truly experimental types, first supersonic, first hypersonic, first swing wing, first tail-less, first VTOL jet, ect. The last few X designations have been the ultra manuevarable X-31, space planes and space crew recoverable vehicles, and, unusually, the fighters X-32 and X-35.
Normally prototype aircraft didn't get the X designation, but rather the Y prefix. When the General Dynamics F-16 and Northrop F-17 (later became the F-18 for a different program) had their fly off for the Air Force lightweight fighter contract, they were designated YF
-16 and YF
-17. The 16 won, and the 17 almost faded into obscurity. Same with the YF
-22 and YF
So, to answer your question, it is my opinion that a university program wouldn't get an X designation and access to Edwards, but I could be wrong. Just everything I can remember that was X had government contracts or armed service contracts behind it.