What would cause my Zoom HN4 not to power up from batteries or mains adapter?

It may be "but" that you're seeing, which indicates the batteries are low. Use the AC adapter made just for this model only, and the guitar needs to be plugged in to turn the unit on. A full manual for this Zoom model can be found here:
https://www.zoom-na.com/sites/default/files/products/downloads/pdfs/E_G1_G1X_3.pdf

It may be "but" that you're seeing, which indicates the batteries are low. Use the AC adapter made just for this model only, and the guitar needs to be plugged in to turn the unit on. A full manual for this Zoom model can be found here:

https://www.zoom-na.com/sites/default/files/products/downloads/pdfs/E_G1_G1X_3.pdf

You need to replace the power resistor R54:
http://gisywall.blogspot.it/2016/01/zoom-g1xn-continuously-shows-bt-on.html

I have a ZOOM GX1XN and it doesn't like universal adapters much. Using one with the wrong voltage or polarity may permanently damage your pedal. Try to obtain a proper ZOOM AC adapter (part number AD-0006 or AD-16) if possible. These are dual voltage and will work equally well with both 220 and 110 volts. Failing that you need an adapter that will give a DC (direct current) output of 9 volts at 0.5amps - maximum 4.5 watts. The outer contact (sleeve) of the adapter needs to be positive (+ve) and the inside contact negative (-ve). Lots of universal adapters are the other way around with the sleeve being negative - but some have a multi-connector on the end with both positive and negative sleeves.

I'm a Zoom 9000 owner (since '93, I think) Did you try batteries only? Are you plugging the guitar into the "in / power" jack and not mistakenly using another jack? When you plug in the guitar, in to the Zoom 9000 does the window come on with the settings shown? Sorry, I know some of these are obvious but always the chance of us being human. If all that seems fine, then I would read though a manual to see if maybe it is a setting problem, i.e., sound somehow got inadvertently turned off. If you don't have one, an internet search will lead you to one. Sorry, wish I knew more.

The first thing to check is the power source. A weak battery or adapter will cause this.
Beyond checking these the only other problems will be internal and require electronic servicing.,

Probably the connector where you plug the AC adapter into has a switch contact that disconnects the battery when external adapter is used and it has gotten bent so when the adapter is not plugged in, it no longer makes contact to the battery power. This is pretty common for devices that use both battery and adapters.

try resoldering the circuit board where the power socket is ... as there is a switch on the socket that disconnects the batterys

Probably two years too late, but I recently got a used H4 and it was having this same issue. Upon reading the documentation, it seemed as though it required a specific type of AA batteries. If you update the firmware (mine was 1.3) to the latest version (currently 2.4), you can toggle the battery type under the menu (most likely alkaline is what you want). It also added lots of other great features!

There is a possibility that the receptacle for the adapter plug has one or more broken solder joints on the circuit board.