Please define French DNA. We are Celt/Roman/Frank/Norman with a dash of Moor, Italian, German, Dutch, English, Spanish, Polish, Swiss...
As a country we invaded and were invaded for two thousand years, not counting our own armies crossing and criss-crossing France during our own internal wars. Then you have the fact that we are a country which for the last two hundred years have had lots of immigrants from all over Europe, and now from other continents. Talk about gene mix.
That said, it's a given that some gypsy women had children with Frenchmen during the Middle Ages some 700 years ago, and even closer to our time, and even during our time. It is also a given that Gypsies in France have French DNA as many have French nationality.
As for what we think of Gypsies, their bad reputation is still pursuing them and making them shunned by the rest of the society, enhanced by the Romanian Gypsies who poured into France and beg and steal in the big cities. And before someone accuses me of being a racist, I had my purse riffled through by a Romanian gypsy kid and fend off the beggars (usually young women and children) regularly so yes, I know what I am talking about. Even the French Gypsies have no love for them.