Is this a false equivalence, or just a reasonable analogy?

This is a reasonable analogy. People tend to call "False Equivalence" when they don't like the fact that the analogy shows their view to be unreasonable. The analogy clearly demonstrates that the mask is not about the individual's health any more than not speeding drunk is.

I see person 1 committing more of a Moving the Goalposts fallacy in their final comment.

Yes, this is a false analogy, one of the most common of all the logical fallacies. It is the same analogy used by gun rights zealots: "cars (hammers, knives) kill people, and we don't outlaw guns."

Person 1: "Wear a mask ...
don't wear a mask ...
it's an individual choice..
The individual's health is their own responsibility.....
When my health becomes your responsibility you're going down a very dangerous road..."

Logical form:

1. Wearing a mask or not wearing a mask is a choice that can only be made by the individual.

2. This is because people need to be responsible for their own health.

3. If people have to be responsible for the health of others, this is a slippery slope that could lead to other undesirable outcomes.

For me, the main issue is premise #2. It ignores the effect of other people's behaviour on one's health. For example, if I carry COVID-19 and enter a crowded area without a mask, I may transmit the virus to another person. My individual health problem thus becomes another person's problem. Furthermore, it is not mutually exclusive to be responsible for your own health and be mindful of the impact of your choices on others. If I wear a mask, it does not follow that others around me can then feel free to, say, not observe social distancing.

#3 thus does not follow from #2. Mask-wearing is a limited-time guideline based on the current public health situation and some scientific evidence. There is no reason to suggest it will set a precedent of forcing people to 'be responsible for other people's health'. One could argue that asking you to wear a mask would be making you responsible for someone else's well-being, although even if true, the degree of responsibility does not compare - in any way - to that of your own health.

"Speed 100mph while drunk or not speed 100mph while drunk during rush hr, its an individual choice. The individuals health is their own responsibility.... When my health becomes your responsibility you're going down a very dangerous road..."

Arguably there is a more clear and present danger from speeding while drunk, although yes, this is a good counterpoint. It illustrates the limits of person 1's "individual choice" argument by showing that some choices may harm others, violating their individual choice as well - I didn't choose to catch the virus from you, and you didn't choose to have your car slammed into by mine.

"that's a false equivalency you're comparing reckless driving on a highway to forcing healthy people to breathe in their own carbon dioxide and follow arrows on grocery store floors." 

I don't see the point of the 'breathing' part...wearing a mask still allows you to breathe normally. No?

Other than that, this isn't a very convincing objection, since not wearing a mask in a crowded area is arguably also reckless behaviour, especially if you are a carrier and in the presence of vulnerable people.

Person 1: "Wear a mask ...
don't wear a mask ...
it's an individual choice..
The individual's health is their own responsibility....

When my health becomes your responsibility you're going down a very dangerous road..."

A mask isn't intended to protect the wearer, it's to stop the spread of the virus onto other people (including via surfaces). We're already going down that very dangerous road.

Person 1: "that's a false equivalency you're comparing reckless driving on a highway to forcing healthy people to breathe in their own carbon dioxide and follow arrows on grocery store floors."

I looked at this recently to check what false equivalence means and it's actually more of a thing to do with comparing things which differ by degree; the correct term to use would be weak analogy in this case. Or not, because it's actually a somewhat fair analogy. 

People are being asked to wear masks for the purpose of not spreading disease, not for the purpose of causing them to breath carbon dioxide , and I've no idea why following arrows is mentioned, when did that become burdensome?

If you want to object to the comparison you could say that people aren't banned from driving because they may be drunk, which would compare with wearing a mask because you may be infected, but that would require a contrarian to actually come up with a reasonable objection rather than just talking garbage. 

Is it a straw man? I'm not sure you could say that as they are the one that started with the false premise of wearing masks for self protection. I think the second person should respond by explaining it rather than expecting them to magically understand just because of the use of an analogy which went over the person's head. 

Not sure I'm seeing a slope either. What would be the thing which gets progressively worse?