The first thing you need to understand is that it is okay to grieve. You have lost someone you loved, and your sadness and pain are natural. Do not let those who can’t understand it shame you into feeling bad about it. They don’t know how special your pet was to you; only you do. You are not immature or overly sentimental, and you’ve got nothing to be ashamed of. What exactly are they accusing you of? Only of having a bigger heart than theirs.
You will probably go through the same stages of grief people go through when they lose a loved one: denial, anger, depression, and maybe some guilt. “If only I did this or that, he might be still alive…” Do not focus on that. More than likely, you have done all you could for your friend.
Another thing I would advise is that you should not grieve alone. Talk to other pet owners, they will understand you better than anyone else. If you don’t feel like talking to anyone in person, Internet is a great solution: there are many pet-related discussion forums you can go to, including some specifically dedicated to pet loss support. Go there and share your story, tell about all those wonderful special moments your friend had brought to your life. Post a picture of him or her for others to see, if you’d like. You will probably cry as you share it all, and then when you read other pet owners’ replies to your post. Crying is good. Letting the pain out is better than locking it up inside.