Sometimes I see people on the internet saying something like "Me has a box" "Me does the job", or "I has a box" (or even "I haz a box"). And I'm sure they do it intentionally.
Why do people refer to themselves using verbs in 3rd person while using "I" or "me" in front of them? And is there any difference between that and using it in a "correct" way (like "I have a box")?
Because I can haz grammar joke.
There's a few different precedents for it. One as Hellion says is the /me command of IRC and other real-time chat and MUD systems.
Another is imitating the speech of young children, to reflect a feigned lack of awareness, or childlike glee.
Another is imitating various characters from sci-fi and fantasy fiction who do not use standard English (Gollem/Smeagol, The Hulk, Yoda, and many "We learn your tongue from your television signals but know it not well though smart we are" aliens).
Another is as part of more general word-play. Bogus foreign or irregular plurals for example are common tech-slang, which is one of the communities the phenomenon came from (boxen is so often the joking plural of box as slang for "computer" as to border on being more jargon than slang, virii was so often used as a joke plural for virus that some non-techies—not getting the joke and thinking the techies must know what they are talking about—thought it was the correct plural, and meeces for mouse is also pretty common [originally copying a cartoon]), doubling verbs for emphasis, modelling English language expressions after the format of computer languages, deliberately wrong choice of suffix (obviousity, winnitude, winnage), self-referential acronyms.
Part of the joke is a self-awareness on the part of the people making these jokes, that they tend to be the very people who get most annoyed by mistakes in grammar. Another part of the joke is they tend to know people who get annoyed even at these jokes, and hence there's some built-in ribbing without even trying.
Me thinks this is just another example of that trend.
One actual "reason" for this behavior is that it is a replication of the command syntax used in many online real-time chat systems. Normally the "/em" command stands for "emote", and "/me" is a synonym due to the possibility of typos. So in a typical online chat system, you'll see things like
Hellion: what's up?
Hellion is bored.
Highstaker shoves a cat into Hellion's pants
That "conversation" is the product of a user named Hellion typing what's up?, and then /me is bored, followed by a user named Highstaker typing /me shoves a cat into Hellion's pants.
