Can a UK national work as a paid shop assistant in the USA?

You cannot do this legally. However, it is quite common for people to work in the US on a cash basis. As there are ~10 million undocumented workers in the US, it is likely that during your travels that you will meet someone who is one of these people.

It is possible that your friend and you are interested in this illegal arrangement. However, the penalty if you are caught is extremely high. There are numerous SE Travel questions & answers about deportation and other penalties that restrict the ability to travel to the US. I think if you are from the UK, it isn’t worth the risk. If you are starving to death, or are a victim of systemic violence in Central America, perhaps it is.

The current very limited possibilities (apart from Marriage to a US Citizen) or as a child of a US Citizen, immigration, include to be a high skilled employee outside the USA for over a year and be so good at the company’s processes that you qualify for an intra company transfer (L1B) as an indispensible specialist knowledge employee transferee. Though shops assistant does not fall anywhere near this category unless you are selling something which it would be exceptionally hard to get a US citizen to be trained at doing. There are further rules effectively barring anyone but larger US corporations from using this route. Another way is to bring lots of money and start a business; invest in the USA and get an E1 Visa. Technically any investment can qualify but realistic chances require at least $100k and leading to the employment of atleast a dozen US citizens, so opening a shop is a better approach. Trump’s new immigration points based system will make it even harder for low skilled immigrants, but easier for high skilled immigrants, especially if younger and highly educated. Watch that space. (Also, Last year there was talk of a UK-US post Brexit work immigration exchange deal which would allow for bi directional labour migration but with current politics that is uncertain where that is going at the moment along with the rest of Brexit.)
Have you considered Australia? They are more open for UK citizens and if <31 you can even get a working holiday visa.

I believe it is too late to apply for 2019, but the J-1 visa summer work travel program allows this sort of work situation if you are a currently-enrolled post-secondary student and your visit is during your academic summer break and within the program dates (which are Jun-1 to Sep-30 for 2019).

Summer Work Travel Program

College and University students enrolled
full time and pursuing studies at post-secondary accredited academic
institutions located outside the United States come to the United
States to share their culture and ideas with people of the United
States through temporary work and travel opportunities.

One needs to apply for this program through a designated sponsor organization.

As the UK is currently a visa waiver country, the sponsor is not required to prearrange employment, as long as applicants can demonstrate they “have sufficient financial resources to support themselves during their search for employment”.

Note that the sponsor organization does not need to be (and is generally not) the employer. The sponsor’s obligation with respect to employment, for applicants from visa waiver countries, ends with providing “participants with a job directory that includes at least as many job listings as the number of participants in their program who are entering the United States without prearranged employment.” Applicants are free to find their own employment, subject to a few exclusions. See the “Program Exclusions” section at the bottom of this page for details.

You would need a non-immigrant temporary worker visa. See for instance Wikipedia

To sponsor such a visa one of the requirements an employer needs to demonstrate is that “there are not enough “able, willing, and qualified U.S. workers” to fulfil their vacancy.

For an unskilled position as an assistent in a shop that is unlikely to be either worth the effort or successful.

The only regular non-immigrant visa category that would conceivably allow working in a shop (no matter whether paid or not) would be H-2B for temporary non-agricultural workers. But that is a purely theoretical option, because it will only be issued if the U.S. Department of Labor finds that it is in the interest of the U.S. economy to let the particular work be done by foreign workers. The chances of this happening for ordinary retail work are nil.

A few other types of non-immigrant visas would allow such work incidentally to the main purpose of the visa, but they all depend on you being in extremely special situations (as one example, victim of human trafficking assisting law enforcement with investigation) or a spouse/child/dependent of the holder of certain other visas. Since you write nothing about such things, that is pretty much a non-starter too.

For completeness, getting an immigrant visa also seems to be impossible. You don’t write that you have any qualifiying family to sponsor a family-based immigrant visa, and the employment-based immigrant categories are even farther removed from retail work than H-2B is. For many, a final (though unlikely) option would be to enter the diversity visa lottery and hope to get lucky, but if you’re born in England, Wales, or Scotland you don’t qualify for that, due to the large number of Brits who settle in the US by other routes.

In short, as the comment said:

Forget about it. There’s no visa for that kind of arrangement.