I d/l Meebo today (for AIM and Yahoo) and I noticed it signs me out too, but I think it signs me out when I've powered the phone off/on. I really haven't thought much about it though until you said something.
Meebo has worked perfectly for AIM, FB and Google talk. That is until today. Today I have logged out or the app force closed about 10 times in 30 minutes. Not sure what has changed. Shutdown and restarted but haven't tested it out again since I am at my computer and dont need IM on my phone.
Meebo would force close on me every time I tried to use it landscape mode, and eBuddy signs me out all the time. (I don't consider an IM client reliable when it signs me out every night, at some point, while I sleep.)
So I'm in the same boat as the OP; I have yet to find a /reliable/ IM app.
I switched to Google Talk though, for now, and it seems to keep me signed in all the time.