Yes. But if your box is one of the original models or a TalkTalk box you'll find the BT issued ones are A LOT faster and more reliable.
Also, BT won't provide any support if you have picture issues etc and they realise you're using a non-BT box.
So I could in theory have my husband sign up to the starter package with the 20 channels for 3 quid but use the humax and be able to record. Also how many Internet channels can be recorded at once I have found conflicting information online?
Yes, you can do that.
But given there's an £49 activation fee taking the first year cost to £85 when you include the channels, why not just pay the £10 per month (no set up fee) and get a brand new box with a warranty and more channels?
Obviously it's ultimately your decision.
Edit: In fact, assuming you already have BT Infinity broadband, you can get the Entertainment Plus package for just £5 instead of the usual £10:
In which case you'd pay a total of £60 - less than the £85 of your suggested option.