all days vs ever day and all persons vs everyone

All days and all persons are grammatically correct phrases which mean what they would appear to mean. But there are a couple things to consider.

First be careful when using persons instead of people. People is more commonly used as a plural of person, unless, as onomatomaniak wrote, you want to sound like a lawyer.

Also, "all days" can't really be used the same as "every day" because the former specifes a set of days and the latter specifies that a singular event occurs on each day. So, for example, you can say

I eat an apple every day.

But you can't say

*I eat an apple all days.

If you want to emphasize that every day, without fail, you will eat an apple, with no exceptions ever, you can say

I eat an apple each and every day.

I eat an apple every single day.

I always eat an apple every day.

All of those phrases add some redundancy to "every day" in order to emphasize that you are meant to take "every day" as a literal statement and not a metaphorical statement.

I get stuck in traffic every day. (Well, not on weekends because I don't drive to work on weekends, and not when I'm on vacation, etc)

I brush my teeth every single day (no exceptions).

Sure you can. Be aware, however, that this construction calls attention to itself and, if employed frequently or weakly, may be more likely to register as an error with your audience.

You'll run into your first example in the phrase today of all days, wherein it's employed for emphasis in just the way you intend.

All persons is seen almost exclusively in legal contexts, where persons is often used in place of people. Instead of that, saying all people may allow you to both use extra emphasis and avoid sounding like a lawyer.