YES — You can apply for a new U2 form to continue receiving your unemployment benefits (Ask the job centre which issued your original U2 form whether you'll have to go back home to apply for this new authorisation or you can do it remotely).
The total period of your stay abroad (i.e. in all countries where you look for work whilst receiving unemployment benefits) may not exceed 3 months (with a possible extension up to 6 months, if the authority paying your benefits agrees).
The answer to this question is in 2 parts:
Your right to stay in another EU country
As an unemployed person looking for work, you may stay in the other country as long as you can provide evidence that you are looking for a job and have a good chance of finding one - copies of job applications, invitations to interviews, positive reactions to your applications, for instance. This is the case even if you have no income or have stopped receiving unemployment benefits.
If you lose your job while living in the other country, you are still entitled to live there and have the same benefits as nationals if you are:
Your right to continue receiving unemployment benefits when you are abroad
You can continue receiving your unemployment benefits while you look for a job in another country for 3 months. You’ll be paid the same amount directly to your bank account in the country where you became unemployed. If you want to stay longer than 3 months, you can apply for a 3-month extension. You must apply directly to the national employment services in the country where you became unemployed.
National authorities are not obliged to grant an extension. They will decide in the light of your personal circumstances. Some authorities grant extensions only in exceptional cases [ndash] for example if you have been ill or had an accident during the first 3 months. Others just require some proof that you are actively looking for work abroad (copies of job applications, for instance).
NO — not at all. You are subject to temporary restrictions on access to the Austrian job market, but this does not affect any of your rights as an EU citizen, including the right to be treated exactly the same as local workers.
If your employers persist with this attitude, you should report them to the Austrian labour authorities.