We are travelling from Porto to San Sebastián in June and would like advise whether there are any direct trains.

No.
Here are your options.
http://www.rome2rio.com/s/Porto/San-Sebasti%C3%A1n

Take a train from Porto to Lisbon, book at www.cp.pt or at the station. Fares from 24 euros if you pre-book.

There's then a direct sleeper train leaving Lisbon around 21:18 arriving San Sebastian at 10:53 next morning.

This train has cosy 4-bed Turista sleepers, 1 and 2 bed Prefrente sleepers with washbasin and 1 or 2 bed Gran Clase sleepers with toilet and shower. It also has cafe-bar.

Fares from just 33 euros with a bed in a shared 4-bed sleeper if you pre-book, a bargain.

Book at either www.renfe.com or (far easier, if you don't mind paying in UK pounds) www.loco2.com

You can fly from Porto to Bilbao and take the bus into San Sebastian if you don't want to spend 12 hours on the train.

You may also board the Sud Express mentiond by man in seat 61 at Coimbra (Porto dp 22.00)
http://www.cp.pt/StaticFiles/Passage..._lusitania.pdf

Forgot to mention the Coimbra option - but as fares are the same as to Lisbon, it's nicer to have dinner in Lisbon and board the sleeper train at it's starting station and get snuggled in bed with a good book and a glass of wine...