I graduated from an US university and got an MS in Japan and then a naitei.
With regards of getting an MS, as long as you get into one of the top universities in Japan (such as TokyoU) it will probably help you. Also, depending on the firm, having a graduate degree means that you will be able to collect bonuses on the first year instead of 2nd year, which can add up to a substantial amount of money for the first year(something like $25k more).
Admissions process: You can just fly over to Japan, hand over the admissions form (usually in July) and take the entrance exam with the Japanese students (in August), if you are accepted, the university will help you get the visa and everything. You don't need an English program for grad school, just email the professor first. If the professor can speak English, you can even go even if the program is in Japanese.
"Anyone know how competitive these programs are?"
As long as you pass their entrance exam, you are good to go.
Regarding JLPT:
OK, as long as you can pass the SPI exams and do interviews in Japanese, you don't really even need JLPT. That being said, the SPI exams does test Japanese, and are sometimes even harder than the N1 level exams. For top firms such as Dentsu, you may need a score of 89%+ to pass the exam phase. (if you don't pass the exam, won't even get an interview. So if you do come to Japan, focus on studying for the SPI exams rather than the JLPT, if you can pass the SPI exams then they will likely never ask you about JLPT)
Getting a job in Japan:
If you actually got an MS in a top Japanese university and can at least speak Japanese, I don't think it's a problem for you especially since you are STEM. Some companies such as Ajinomoto actually have foreigner quotas that their HR must fill (something like 10% of new grad hires). On your first year, you should sign up for all the job hunting sites: rikunabi, mynavi, goodfind etc.
and with regards to transferring into a Japanese company after work experience?
Next to impossible especially if you don't have native level Japanese. Japanese companies hires most people as new grads.