Should non-United States residents pay taxes upon receiving stock from a U.S. startup they have worked for remotely?

Likely your best avenue is to call H&R Block Executive Tax services.

There might be one in your country.

If not, just pick one from any US City. I'm in Austin, Texas + have been using them for years.

They're way cheaper than a CPA + tend to stay current on new tax code, far better than a CPA, as they run internal, company wide trainings, all year round.

You're smart to deal with this up front.

Avoiding US I.R.S. trouble == a very good idea.

If you call any accountant or small business bookkeeper you will most likely receive an answer through a quick phone call. You also try asking this question on Quora.com - the people that work at HR block (a tax preparation franchise) don't typically know more than what the computer walk-through provides them, they are mostly temp jobs and trained on the job for the tax season mostly. Your best bet is to go to Quora.com and ask. I'm not a certified accountant but I did take accounting classes during my MBA program and have been working with startups and companies in various countries... here is my answer to your question:
I hope it’s not confusing:

Non-residents are subject to U.S. income tax only on U.S. source income, they are subject to two different tax rates. One for effectively connected income (such as wages, self-employment earned in the U.S. or from the operation of business in the U.S.) and is taxed the same way as a U.S. resident. The other taxation is for income that comes from passive income such as interest, dividends, rents, or royalties, this income is taxed at a flat rate of 30%, unless a tax treaty specifies a lower rate. Non-Residents must file and pay taxes using Form 1040NR.

So to answer the question, YES you will have to pay taxes on stock shares received from company #2 since it’s registered in the USA and the income is being reported in the USA.

My name is Humberto Valle, I'm a marketing strategist for http://Unthink.me - we bring big business tools and experts to small startup growing companies. Our facebook page is facebook.com/iwillunthink - I would really appreciate it if you follow me there ;)

You and your company both are two separate entities. Your company is in US, therefore it has to pay taxes in US. Therefore, when you receive the stocks, you don't have to pay any taxes in US as your company is already paying taxes in US.
As you are staying in Argentina, you are using the public resources like road, governance, security (police), justice, etc; it makes you liable to pay taxes there. You have to pay taxes on your income as per the tax rate and slabs that you fall in.

Thanks for reading
Shishir