Start by considering your friend's position
If you accept the premise that there is systemic racism making the lives of certain people more difficult through no reason beyond their heritage or colour of their skin (and it sounds like you do as you want to support your friend). Then I suspect the difference of opinion between 'Black Lives Matter' (BLM) and 'All Lives Matter' (ALM) is merely misunderstanding the sentiment.
BLM is coming from a position of being treated as if they don't matter. It is people standing up and saying that, despite how they have been treated, they do in fact matter. ALM is coming from people who, having never been treated as though they don't matter, are (ironically) insisting that no one should get special treatment. All BLM is asking for is the same treatment as others.
Your key issue seems to come from:
To be blunt, I'm afraid my friend is saying "if you're not with us you're against us" (he's not specifically using those words, but I've seen other people use them).
I can understand feeling uncomfortable with this. I know that what you want is to not get involved - your life is simpler that way. The unfortunate truth, however, is that silence only supports the status quo*. If you don't treat the issue as important you'll only contribute to it's persistence.
You can't not be involved in the politics when even your silence is a statement. It does put us in a difficult situation - I don't like posting about politics on social media either but after a great deal of thought I came to the conclusion we have no other option. I'll put details of my experience (as is required) at the end.
What next for you
The key points from what you've told us are that this is a good friend and that you want to be supportive.
Before starting the conversation ensure that you've accepted the fact that this is something you've no experience of. You need to listen to them - pretending you've knowledge about how they're experiencing this is one of the key problems here.
That being said I would start with talking to your friend openly about this and understand their experiences of racism (both open and systematic). Avoid using phrases like BLM and ALM for now because I doubt you'll find your differences are more than semantics.
Your friend is likely on the defensive - steer away from putting your own opinion forwards until you've considered all they have to say. Be willing to change your mind.
A good starting point would be something simple like:
Hi [friend], I've seen your posts recently and I'll admit I'm struggling with some aspects of this. Can I get a bit more of your perspective?
Another thing to consider is whether you call or talk over private messages. Messages give you time to consider what has been written, reread if necessary, and give a more considered response but what they don't carry is the emotion behind the words. A question you've asked in earnest can sound accusatory if read wrong, this is where a phone (or video) call could be more beneficial.
This is a good friend - you don't want to fall out just because you've misunderstood what a phrase means.
My personal experience is in talking to my own black friends after seeing a rather unhealthy argument break out on social media. Before really thinking about it I had considered BLM as divisive, putting some line between black and white people.
I started a conversation similar to the one I've suggested and after a fair bit of talking (over the phone in my case) I realised that BLM isn't putting the line between races, it's highlighting one that is already there. A line that has been slowly broken down to a point where is no longer visible in the form of separate water fountains, toilets and bars but still exists as being less likely to be called for a job and more likely to be shot by police.
*Silence being equal to assent (and this the persistence of the status quo) isn't a novel idea. It is regularly implemented in the EU and law.