Your question seems a bit confusing. What do you mean exactly? Maybe:

He asked me if I would mind working on Saturdays.
He asked me if working on Saturdays made me happy.

He asked me if I was happy to work on Saturdays = Er fragte mich, ob ich damit zufrieden sei, samstags zu arbeiten.

He asked me if I were happy to work on Saturdays = Er fragte mich, ob ich damit zufrieden wäre, samstags zu arbeiten.

Unterschiedliche Bedeutungen also: Im ersten Fall arbeitet die befragte Person bereits samstags, im zweiten wird nach der Bereitschaft, das zu tun, gefragt.

Nein, Werner, das ist falsch. Im ersten Falls müsste es heissen: He asked me if I was happy working (on) Saturdays.

Im zweiten Fall müsste es (IMO) lauten: ... if I would be happy to work ...

Possible would be: He asked me if I was happy to be working at all (or on Saturdays).

Es handelt sich hierbei um eine Grammatikübung: die Frage "Are you happy to work on Saturdays?" soll in Reported speech (Past tense) umgewandelt werden. Ist bloß die Frage, ob sowohl "was" als auch "were" korrekt sind???

He asked me if I was happy to work on Saturdays.

This is a perfectly reasonable question in BE.

Ist bloß die Frage, ob sowohl "was" als auch "were" korrekt sind???

Was is correct.

He asked me if I was happy to work on Saturdays.

This is a perfectly reasonable sentence in BE. I don't think you could use "were" in standard speech, but I'm not 100% sure.

'Were' is wrong, sorry. Perhaps you were thinking of 'would be.'

I would also suggest 'willing' instead of 'happy,' or 'mind' instead of 'be' plus an adjective.

He asked me if I was willing to work Saturdays.
He asked me if I would be willing to work Saturdays.
He asked me if I minded working Saturdays.
He asked me if I would mind working Saturdays.

Then you can use 'happy' or 'glad' in the answer:

I said I was happy / would be happy to.
I said I wouldn't mind in the least, in fact, I'd be happy to.

"Are you happy to work on Saturdays" sounds jarring to me. I think an AE speaker would instead say something like "Are you happy that you work on Saturdays" or "... happy to be working on Saturdays." Maybe it's the seemingly contradictory juxtaposition of "happy" and "work on Saturdays" that's doing this to me. :-)

Interesting - I take it that in AE you just wouldn't ask someone, "Are you happy to work at the weekend?" It's perfectly normal in conversational BE.

for one thing, I'd say "work on weekends" in AE. But also, I'd use the progressive: Are you happy to be working ...

That's a different question, though. "Are you happy to work Saturdays" would mean "Are you willing?", not "Do you like it?"

Then maybe that's the "problem." It wouldn't mean that in AE.

Rats, hard to edit around here ...

The difference to me is that 'happy' is asking about the present, what you're already doing, whereas 'willing' is about the future, a hypothetical condition.

Are you happy working weekends (= as you're now doing)?
vs.
Would you be willing to work weekends / this weekend?
Would you be happier working weekends?

I think that's more or less what dude is saying too, so maybe there is some AE/BE difference (besides 'at' vs. 'on' the weekend).

I think what is meant by "Are you happy to work Saturdays?" depends (surprise, surprise) on context: in a job interview it would mean "Would you be willing to work on Saturdays?". If I'm telling somebody about my irregular working schedule they might ask "Are you happy to work on Saturdays?"

I agree with Capt. Flint and mikefm re "happy" in BE.

@ pokito_1

As others have said, were is wrong.

You are probably confusing the irrealis (so-called past subjunctive) were, which (in formal English) is used in counterfactual conditions, with the preterite was, which is the "backshifted" form used in reported speech.

In English, reported speech uses the preterite (so-called "simple past") where German uses the subjunctive. You may be confused because it follows "if", but it is not a conditional clause.

Compare your example with the conditional clause in "If I were happy to work on Saturdays, I wouldn't ask for more money.

Man: "Are you happy to work on Saturdays?" (Is it OK if we call you in then?)
He asked me if I was happy to work on Saturdays.

Man: "Would you be happy to work on Saturdays?" (Would it be OK?)
He asked me if I would be happy to work on Saturdays.

You are confusing two varieties of 'if'.

There is the conditional 'if', and there is the indirect question 'if' (= whether).

Here you have the latter, so the subjunctive 'were' does not apply and the usual rules of indirect speech do apply.

He asked: "Are you happy to..."

He asked if I was happy to...