Refused to enter the UK, will Amsterdam let them in for an overnight stay whist waiting for a flight back to Orginal Country, Argentina

Per TIMATIC, the database used by airlines:

Passengers are not allowed to enter.

This does not apply to passengers arriving from Austria, Belgium, Bulgaria, Croatia, Cyprus, Czechia, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland (Rep.), Italy, Latvia, Lithuania, Liechtenstein, Luxembourg, Malta, Norway, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden, Switzerland or the United Kingdom.

Unclear, however, if being sent back from the UK qualifies as arriving from the UK.

I'd recommend you to email Dutch border control on this, at ltc.opscent.kcc@mindef.nl. Write "Urgent!" in the message title

The title implies (but doesn't state directly) that the boyfriend flew from Argentina to the UK. Arriving in the UK, he was denied entry.

Because he's now detained, UK Border Force will want the boyfriend to depart for Argentina, where UKBF is confident he'll be admitted. Because the boyfriend was found not admissible to the UK, UKBF will not release him to spend the night in the UK or in the airport. On these facts, UKBF will similarly not release him to fly to an uncertain acceptance in the Netherlands.

If the Dutch authorities, however, assure UKBF that the boyfriend will be allowed entry into the Netherlands, and the boyfriend can secure a ticket from the UK to the Netherlands on short notice, then UKBR might allow him to fly to the Netherlands.

I agree with Crazydre's recommendation: all you can do at this point is to request the Dutch immigration authorities to transmit such assurance to UKBF. You could also ask UKBF to make the same inquiry of the Dutch.

Detention facilities within airport terminals are small and limited. If UKFB can't quickly find another country to receive the traveler, they'll likely move him to a UK immigration detention center in no more than a day or two.