Is "you Chromebook" correct grammar in any dialect?

You are correct, whatever is trying to be said here, it is not standard grammar in any variety of English. But...

In a long tradition of being loose with language, the people who created the advertisement for Google are using words in a new way to make the message memorable. In short, they are attempting to verbify the noun 'Chromebook'.

To explain, the context of the video is a number of people saying "I'm dying" when using their phone, notepad, laptop, anything that traditionally runs out of battery at inopportune moments, also involving the uncomfortable ways to deal with that at the last minute. Before a meeting running through the halls, in the airport squeezing behind the seats to find a plug, unravelling messy cords.

The solution for all this dying, given at the very end of the ad, is

If you're tired of dying, You Chromebook.

This wording leads one to parse the sentence as 'You (verb)'.

Analogous sentences are:

If you don't like the heat in the kitchen, you leave.

or closer to the subject (and verbifying a noun)

If you don't know something, you google for it.

Verbifying a noun (sometimes said 'verbing a noun', using a noun in a verb position), is often seen as a solecism (an error) i standard English, but it is commonly done and is the source of lots of new terms once people get past the informality. The classic example is 'gifting' for 'to give something as a gift':

I gifted my aunt a turquoise vase, but she regifted it to my cousin".

It is very common to informally verbify a noun, but is mostly an informal method of neologism. Many (older?) people think of this as 'wrong', but then again it is very popular. On a case by case basis will any particular instance sound acceptable (usually based on how commonly it is used).

The construction here is a bit infelicitous. 'Chromebook' is an ungainly to shoehorn in as a verb. What does that really mean to 'chromebook' something, use a Chromebook? That's pretty underwhelming. I suppose they intend people to understand it as 'You buy a Chromebook to avoid dying so often'. But for me it just doesn't work, it doesn't have a natural interpretation as an action.

That explains the most appropriate interpretation of the situation "What does this really mean?" because it is not obvious at all. But to more directly answer your question, no, the advertisement is not trying to be hip by using a non-standard dialect; instead they are trying to be hip by using a non-standard pattern used by many speakers of the standard version). I don't know of any varieties of English where 'you' is used as the possessive pronoun, eg instead of 'your book', they use 'you book'. Maybe some obscure pidgin.

Two notes about the production of the ad:

• The many times repeated sentence "I'm dying" is an attempt at humor also. They're not literally dying, losing their life, but rather 'dying' is a common metaphor for losing battery life. Older parents are calling adult children from their retirement beach and notice that they're losing power and then haha they say they're dying right before the phone call might cut out. Uncomfortable death humor.
• The production has problems. All the 'joke' instances about dying seem to build up to a big joke about dying at the end, but instead. That dude is looking up at the woman in an uncomfortable manner while plugging in to a socket on the floor. The capital 'Y' in 'You' is out of place and misleading. A lot of these details are intended to be new and thought provoking but they're just confusing and taking away from the point of the advertisement.

"You Chromebook" is grammatical in all standard English dialects, as long as you accept that Chromebook is being used as a verb.

I don't know what the verb Chromebook means - I assume it means to use a Chromebook brand computer - but that doesn't make the sentence ungrammatical.

Not only is this sentence grammatical, it's relatively common to turn a brand name into a verb. Consider:

"If you need copies, you Xerox"
"If you want to have fun on the lake, you Jet Ski"
"If your back is sore, you Jacuzzi"
"If you like to go fast, you Rollerblade"
"If you don't want to walk, you Uber"
"If you need to see the person you're talking to, you Skype"
"If the photo needs to be touched up, you Photoshop it"
"If you don't have cash, you Venmo/PayPal"
"If you're tired of your toys rolling away, you yo-yo"
"If you're building a house, you Sheetrock"
"If the package needs to get there fast, you FedEx it"

And on and on and on