they say an Airport Express will not connect with a non-Apple router.
Not correct. I helped my neighbor set up his two AirPort Express devices with an Orbi wireless network using a normal wireless connection and things work just fine. The Expresses are set up to "join a wireless network", like I mentioned above. No wired or Ethernet connection is required.
Here, I am assuming that you have a recent version of the AirPort Express.......the one with two Ethernet ports (Model 1392), or an older model....the one with one Ethernet port (Model 1264).
My son has a Fios wireless network and has configured his old AirPort Express to "join" the Fios network as well. No problems at all.
In the example below, you can see an AirPort Express that is being set up to join a wireless network provided by an AT&T 2Wire wireless router.

What the Apple Support folks might be saying is that the AirPort Express cannot wirelessly "extend" a third-party router and provide extended wireless coverage......like it can with another Apple router.
Let us know what device.......a Mac, an iPhone / iPad, or PC that you are using to try to configure the Express, and we'll provide a few more tips that might help.