Rogers opened the scoring in England's 2-1 UEFA Nations League victory over Croatia on Wednesday, with Harry Kane also finding the net.
Palmer was left out of Thomas Tuchel's squad after the England manager raised concerns about his limited opportunities to impress.
The 24-year-old, who also missed out on World Cup selection, withdrew from September's international camp through injury.
However, Rogers was quick to defend his childhood friend, firmly rejecting claims that Palmer lacks commitment to the national team.
"He wants to be here,” Rogers told BBC Radio 5 Live.
"He wanted to get called up and he wanted to come and represent his country as he always has done.
"He was in quite a low place with his injuries last year and didn't feel 100% himself on the pitch, so he wants to be here and he wants to be the best version of himself.
"But he also has to look at his body. He wants to play for 10 or 15 years, he also has to manage himself. If he has to manage himself in the short term for the greater good then he has to do that and you have to fully support it.
"It has nothing to do with England or not wanting to play with England. It's a nonsense really that people are questioning someone that wants to play for England. It's a lie.
"You see the joy and smile on his face, how much he loves the game, so I honestly don't get it and he doesn't get it."