They could be using the old HR blackmail trick from the book where if you agree to leave they promise to give you an awesome letter of recommendation whereas if you let them fire you for underperforming reasons they'll give you a iffy sounding letter of recommendation.

In Portugal being laid off means (aproximately) getting 2 thirds of your sallary (one third payed by state and one third payed by the employer) and hope to get back to work. Being fired means getting 80%(?) of your sallary and possibly a severance IF you have worked continuosly for at least 6 months AND have a dependent work contract (or equivalent). If you quit you get nothing.

You are confusing a layoff and a furlough. It's not your fault though, the Portuguese government decided to call a furlough a layoff for who knows what reason.

Don't know for Portugal, but in Belgium "underperforming" is also an economic reason, you get the social security protection. Misconduct on the other hand gets you suspended from govt protection for some times.

Unless you do something so horrible that it is talked in the news/wide industry, rarely any company will contact your previous employers. They may ask 'so, why did you leave company X?' but in 20 or so years, I never had a company in Portugal ask for an actual reference. (I had in the UK, US and Switzerland though)

While I am not sure, I think legally here is a very grey area as employment information is protected, but I'm not 100% sure.