Can a resident of Chutz L’aretz only keep one day of Yom Tov in Eretz Yisroel? I heard that is the Lubavitch minhag. What is the basis of residents of Chutz L’artz having to keep two-days on EY?
Yes, depending on a number of factors. Ask your LOR.
Generally single (unmarried) sefardim keep one day, and the minhag among Ashkenazim is those who have a chance of returning to chutz la’aretz hold 2 days – I have been living here for 2 years now and we keep 2 days. There are different factors involved, and each person should ask their LOR, but this is the generally prevalent minhag AFAIK. For a lengthier discussion there’s a Sefer called Yom Tov Sheini K’hilchoso.
Does this work in reverse, whereas there are shittas that residents of Eretz Yisroel keep two-day Yom Tov when in Chutz Laretz?
There is a concept of 1 1/2 days YT. It is a little tricky, but basically we’re mechuyav in the lo sa’asehs but potur from the asehs, meaning we can’t do melocho but we don’t have to make kiddush. I believe there are some opinions that a person from EY could do melocho on the second day of YT in private, but I’m not 100% on that.