So I couldn't find a direct answer to this problem in my Google searches and then came across this thread.
Basically, I also had a Samsung Galaxy S3 and switched to the iPhone 6+ in December. I had been using Google Voice for a few years on Android without incident, but when I switched to iPhone, I was getting some texts sent to iMessage and some to the Google Voice app on iOS. It was frustrating, so I deleted the GV app while disabling the service altogether. Since then, 3 months later, I have not been able to send/receive text messages from non-iPhone phones and didn't know why.
I tried disabling iMessage, restarting iPhone, and re-enabling it, but nothing! I then saw this thread and had the wild idea of enabling GV service again, while installing the app to iOS. That seemed to work! I can now send and receive texts from non-iPhone devices, as well as Android and everyone else!
The only things I don't like about having two messaging clients is that some texts get sent to iMessage and some to Google Voice, but both are never synced with one another.
Aside from that, my problems are fixed.