A generic Home button should work fine.
The obvious question is whether the replacement part you ordered is good. Small parts are notoriously hit or miss. Did you order it from a reputable vendor or eBay/AMZN?
Assuming your button is good, I would check the following:
* Is the bracket properly installed (see step 19 of the [guide|29706|iPhone 6+ Home Button guide])?
* Is the flex or button damaged in any way?
* Check the FPC connectors between the Home Button flex and the Extension flex that goes behind the back-plate (see step 22). The connectors may be improperly seated or damaged as they are quite delicate.