This (Norwegian vegan body builder!) dude would say 'nei!'
I can't answer your question definitively, as I'm still learning Norwegian myself, but the impression I'm getting about the difference between behøver, må and trenger is this:
• å behøve is used to denote when something would be good, healthy, or beneficial for the subject of the sentence. I think it's also an auxiliary verb, to be followed by another verb in the infinitive form: "Behøver jeg å drikker melk?" = 'Do I need to (is it good for me to) drink milk?'
• å måtte (må) is used when something is required by the rules or by an authority, and is followed by another verb in the infinitive form, but without the preceding å: "Må jeg gå på skolen?" = 'Must I/do I have to go to school?'
• å trenge is used with a noun (i.e., not with other verbs) to show when an object is needed or physically required: "Vi trenger tre egg for å lage kaken" = 'I need three eggs to make the cake.'
If a native or fluent speaker could confirm which parts of this I have got right - or if I'm wrong, and some of the words are completely interchangeable - I would very much appreciate it!