How do I handle getting the silent treatment from my boyfriend?

You are in an abusive relationship-not because of your boyfriend's silences but because he is comfortable being sarcastic and hurtful, never admits he is wrong and never apologizes. It is your choice to participate in an abusive relationship. You can not stop an abusive partner from being abusive no matter how wonderfully you treat your partner. You can either accept the terrible relationship as it is or reach out for the support that you need to leave the abusive relationship. I wish you well, always.

This has to be very difficult to endure.

It's pretty common for members of a couple to have different ways of resolving conflict. Very often, one person wants to deal with the conflict right away or shortly thereafter and the other person wants to wait. One thing that sometimes works is to consider having a conversation when you are not upset with one another about taking some sort of timeout and setting a timeframe for something like "I cannot talk about this now, but we can talk about it in two hours."

My best guess is that your boyfriend is triggered by some previous relationship, either romantic or in childhood. If he is aware of something that he thinks or feels when you ask him to resolve conflict or to discuss it, it would be good to know what that is. It could even be that he's learned to handle conflict that way.

Something else you could try is to have a conversation where you each discuss what is acceptable for what is acceptable for each of you during an argument and what is not. These sorts of discussions can be very difficult and you may benefit from working through them with a mental health professional who has some experience working with couples.

As far as what you can do now, perhaps you can remind yourself that these things have worked out before, that you love him and believe that he loves you, or some other statements that are true for you. You may also consider finding ways to soothe yourself. For example, you may find that you like certain outdoor activities or reading a certain type of book or something else that gives you pleasure.

Try during a good moment between you to tell him you'd like to talk over something about your relationship and that you'd like to set a time with him to do so.

When a person shuts down and withdraws, the only choice is to withdraw yourself.   If this continues on a regular basis, after a while the two of you will become increasingly distant and have developed the very unhappy habit of not clearing the air between the two of you.

Good for you to recognize a problem and be willing to address it!

 This sounds very hurtful for you to be on the receiving end of this. You can ask yourself " what am I learning from the way I am being treated? "  and consider whether or not this is in fact either (1)  A message or value that you agree with and believe will strengthen your relationship and help you to grow as a person or (2)  A message or value that is damaging to the relationship or  to your view of your self and others.  Once you've considered that and come to your own conclusion, you will likely know what you need to do.  If you're still stuck, you may want to consider seeing an individual therapist for yourself  to process your feelings about the relationship or a couples therapist with your partner to work on improving your communication with each other.