There is a brother in my community who greets other people by physically embracing (hugging) first and then makes Musaafaha (shaking hands). Then there are other people who shake hands first and then embrace. The question is, is there any evidence that the Prophet embraced first and then shook hands?
Hugging and handshaking are two different things. Hugging is called "Muanaqah" in Arabic and handshaking is called "Musafaha".
Musafah is proven to be very common in the life of the Holy Prophet Sallallahu Alaihe Wa Sallam while hugging would be practiced upon returning from a journey or with a visitor from other cities etc...
When someone wants to practice hugging and handshaking both then there is no specific order. One may shake hand first and then hug or one may hug first and then shake hand. Although the more common practice is to hug first and then shake hand.
And Allah knows best.
Mufti Ikram ul Haq
Fatwa Center of America