Could a state prevent its residents from having abortions in other states/countries?
There is an unenumerated constitutional "right to travel" (which has been recognized in case law, and has not yet been judicially overruled) and there is also a concept called the "dormant commerce clause" which prohibits legislation by a state that interferes with the ability of people to engage in interstate commerce even if Congress has passed no relevant legislation.
There could also be a privileges and immunities clause argument arising under the original 1789 constitution and not the 14th Amendment to that document privileges and immunities clause, which affords people from outside a state the same rights as people in a state.
Also, citizens of a state are defined as its residents, so a state only has jurisdiction over someone as a citizen for so long as they reside there. I was born in Georgia, for example, but haven't lived there since I was six years old, so I am not a citizen of Georgia.
The proposed Texas law bears some similarity to the Mann Act of 1910 which prohibits transporting people across state lines for the purposes of prostitution (to slightly oversimplify). But the Mann Act is a federal law, not a state law.
The proposed Texas law also bears some similarity to the infamous Fugitive Slave Act of 1850 which required free states to respect the slave status of people treated as slaves in a slave state under the slave state's law by returning fugitive slaves to their out of state masters, when the slave escaped across state lines. But, this was also a federal law and reflect the greater extraterritorial force of contracts and property rights created under state law compared to the extraterritorial force of the police powers of a state government.
There are constitutional provisions requiring states to honor each other's rulings as well, most notably the requirement to extradite felons, and the full faith and credit clause that requires states to honor the court judgments and government determinations of status (e.g. marriage certificates) of other states in most circumstances. But, I don't think that you get there in a case regulating the conduct of a state resident outside the state, or in a case where you want to criminalize assisting someone in the state to leave the state for a particular purpose.
Neither of these examples, however, involve state laws. Generally, penalties for doing something across state lines need to be established by federal, rather than state, laws.
This said, the issue has not been litigated in this particular context yet, and the legal theories implicated and structure of those lines in fine particulars could matter. States have only rarely tried to regulate the conduct of their residents outside their own states and have even less frequently been successful in doing so.
First, citizens of their states would be more accurate.
I am certain the commerce clause would nix such a scheme. Furthermore, commerce clause might also come into play to "force" health care companies operating in states that have set strict limits on abortion to provide for out-of-state abortions.
The wide interpretation of the scope of the Commerce Clause continued following the passing of the Civil Rights Act of 1964, which aimed to prevent business from discriminating against black customers. The Supreme Court issued several opinions supporting that use of the Commerce Clause. Heart of Atlanta Motel v. United States, 379 U.S. 241 (1964), ruled that Congress could regulate a business that served mostly interstate travelers. Daniel v. Paul, 395 U.S. 298 (1969), ruled that the federal government could regulate a recreational facility because three of the four items sold at its snack bar were purchased from outside the state.
Commerce Clause is a great big hammer that has been used in the past to destroy state rights by ideologues on the right and left. Don't see why it wont be used this time as well.