yosemite keeps pushing external drives out

So to recap:

You have an external hard drive connected to your iMac via USB and it disconnects at random?

If that is the case and you have a new iMac under warranty, then I'd suggest contacting Apple support or taking it to your nearest Genius Bar. I purchased a new iMac and it was doing the same thing > possible suspects were bad USB ports/logic board/power management problems. It was suggested that I return the iMac as DOA (I was within the 14 day return period) and got another one which has not exhibited those problems.

Last week t replaced a defunct 2007 Seagate 80GB Ext Hard Drive (EHDD) USB'd to my Mac 21.5. I did this with a new Toshiba Canvio 1T Slim II EHDD for Mac plus a Toshibe Canvio wireless USB Adapter Purchased direct online from Toshiba.com. Installation required the built in software in the EHDD. Toshiba has good tech support. The app for portable devices and the Mac is Pogoplug, as recommended by Toshiba. All that being said, contact your EHDD manufacturer for support to Install. After that go into theMac Settings/time machine, open Time Machine and select On. You can set the timer for frequency for TM to store your stuff on the EHDD. I hope this helps and good luck.

Not sure why you are addressing me or what it has to do with the OP's question, however..... it is generally not recommended to use any software installed on any external drive as it is not needed. TM is far more reliable with a wired connection. And, the OP did not ask about icloud backup.