So I purchased a lightly used Epi ES-339 from Guitar Center a few months ago, and have been having a great time playing it (only been playing for a year or so). Everything was great with the guitar itself, except the one volume pot had a tendency to slip just beyond the "10" indicator, when light pressure is applied. I took it into guitar center, and they said they would repair for free, but it would take a few weeks to get the part specifically from epiphone. I have a great little guitar/music store new my house, could they just replace the pot using another brand pot? I know I'm knew but I can't imagine there's too much difference between the brands of a component. It plays perfectly fine with the faulty knob for now, I'm just a bit of a perfectionist with some things lol.

You can absolutely replace with another brand. The relevant specs here are the size (which is standard for guitar pots), the resistance (probably 500K ohms, very common) and the taper (audio taper, also very common). 

In fact, most aftermarket brands are going to have a higher quality pot than the OEM Epiphone one. A lot of people replace the stock ones for a smoother, more reliable model. 
Thank you. My only other concern is that the epiphone volume pots can "pull up" for a different tone, would this be possible?
A pot that turns past 10 usually has a loose knob or the stem nut has loosened up. I've never seen the knob turn too far because the pot itself is broken. Not to say it's impossible, just extremely unlikely.  Did a tech actually examine the guitar or was it just a salesperson giving you a stock answer? 

Anyway, the pot doesn't have to be an Epiphone brand part...in fact, it almost certainly is a generic brand anyway. The important issue is the resistance value (which is probably 500k ohms)  taper, and physical size of the pot.  The value is stamped on the bottom of the pot, and sizes are pretty uniform. I would be amazed if Guitar Center doesn't have generic replacement pots for sale on a rack somewhere in the store.  Any guitar tech (or yahoo with a soldering iron) can switch out one of those babies in a matter of minutes.