Yes you will, Yosemite will be just fine. The problem with all the later digital download operating systems is ownership always belongs to the Apple ID that downloaded it. Should you continue to use Yosemite you would not be able to get Security and General Updates which are vital with so much malware about.
When you get the Mini, get Yosemite and do as suggested. When up and running download AdwareMedic, AdBlockPlus and Ghostery to control all these nasties on the 'net. Never download anything from softonc or CNET as they are noted for this.
The Mini you watched Yosemite on could have had insufficient memory, a full hard drive and not had Onyx run over it, the best cleaning app available for Macs. Also there is a Firmware Update for that Mini, version 1.2.