Is It Prohibited to Accept Birthday Gifts?
Question

I do not celebrate my birthday but if I am given a gift on my birthday, what should I do? Should I accept it or refuse it?
Please note that my friend may get upset with me if I do not accept her gift.
Usually my family celebrates on such occasions by making sweets. What should I do? Is it permissible for me to eat from them?
What if I am given a gift one day after my birthday, and I know that it was bought because of this occasion and for no other reason? Is it permissible for me to accept it?
Praise be to Allah.Do Muslims celebrate birthdays?
Celebrating a person’s birthday and taking it as an “`Eid” or festival that comes every year is an innovation and imitation of the non-Muslims . Hence it is prohibited to celebrate it whether that is done as an act of worship or as a custom and tradition. 
You have done well by not celebrating it, and we ask Allah to reward you with good for that. 
If your family celebrates that , then you have to do two things: 

Offer them sincere advice and explain that this is something that is not allowed in our religion.
Do not take part in the celebrations and avoid doing anything that could be understood as approval.

These guidelines will help you with regard to gifts that are given to you and with regard to eating food or sweets that are made on these occasions. 
Is it prohibited to accept birthday gifts?
The basic principle is that you should not accept any kind of gifts that come on such occasions, whether they are given to you on the same day or afterwards, because accepting them implies approval of the occasion and helps to perpetuate it. So you should excuse yourself from accepting them, with the best of manners. 
If you are afraid that it will spoil the relationship between you and your friend, then explain to her that you are accepting this gift because of her friendship, not because of this innovation, and explain to her that you will not accept that in the future and you will not give her a gift on her birthday. 
Is eating birthday cake prohibited?
With regard to food or sweets that are made for this occasion , you should definitely avoid them, because making and eating them are part of the celebration ; not eating it means that you are clearly objecting to this innovation, and that may be a cause of their giving it up. 
And Allah knows best.