Another poster just within the last few days described how his pressed-in zerks came out stuck in the grease gun nozzle when he greased. I have also had pressed in fittings pop out from grease pressure beneath them.
Grease fittings are bolt thread sizes and 1/8-inch NPT pipe. And metric sizes along with 1/8-in BST British pipe. Refer to Alemite or Lincoln catalog online for sizes, data, and general info.
Clean the fitting hole really well (dental pick, Q-tips, solvent, flashlight) and make certain you have threads and then try known thread sizes--using a selection of zerks--until you get a fit. Don't booger the threads. If you snug it in and holds when grease applied it's the right thread.
If is a pressed zerk, you probably won't get a replacement to hold. You need grease. Start thinking about disassembly and tapping the hole. If you tap without taking the mechanism apart you will probably introduce metal flakes deeper into the hole around the bearing / shaft and make your problem worse.
I have reinstalled a press-fit zerk by using a tiny chisel and slightly deforming the internal lip with a burr that would hold inside the hole. Never liked doing it but sometimes seemed the only reasonable way. Much better to thread the hole.
Press-in type zerks come in two varieties: (1) pop right out and (2) can't ever get out.
Good luck and please post back with your experiences so we may all learn.