John Smith doesn't care about voting. Is it legal for him to give his ballot to Jane Doe and let her vote, then have her give the ballot back to him so he can sign and submit it? Does it matter if Jane is eligible to vote or not?
At a minimum, what you describe is a violation of ORS 254.470(6)(a):
Upon receipt of any ballot described in this section, the elector shall mark the ballot, sign the return identification envelope supplied with the ballot and comply with the instructions provided with the ballot.
With one exception, only the voter is permitted to mark their ballot. That exception is ORS 254.445(1):
If an elector is within the county and, because of a physical disability or an inability to read or write, is unable to mark the ballot, the elector may request and shall receive the assistance of two persons of different parties provided by the clerk or of some other person chosen by the elector in marking the ballot.
It may also constitute perjury: the procedure for returning a mail ballot in Oregon involves signing a statement on the return envelope. The wording of that statement is not prescribed by law, but in some counties, may include a clause to the effect of "I filled this ballot out myself". This would be a violation of ORS 260.715(1), which is a class C felony.
A person may not knowingly make a false statement, oath or affidavit when a statement, oath or affidavit is required under the election laws.
If "Jane Doe" has already voted once in the election, it is a violation of ORS 260.715(3), again a class C felony:
A person may not vote or attempt to vote more than once at any election held on the same date.
In the state of Oregon, voters who are unable to fill out their own ballots may request assistance filling out their ballot. (ORS 254.445, emphasis added).
If an elector is within the county and, because of a physical disability or an inability to read or write, is unable to mark the ballot, the elector may request and shall receive the assistance of two persons of different parties provided by the clerk or of some other person chosen by the elector in marking the ballot. The persons assisting the elector shall ascertain the wishes of the elector and assist the elector in voting the ballot accordingly, and thereafter may give no information regarding the vote.
I have the impression, based on anecdotal evidence, that “some other person chosen by the elector¨ typically means their caregiver, and that this is much more common than requesting that the county send over a pair of strangers from different parties.
A disabled voter may also be allowed to use a signature stamp, or if necessary authorize someone else to use the stamp on their behalf.
There is one other half-exception, described as election workers in the furtherance of their duties. What this means is: please don’t use white-out on your ballot! If you do, the optical scan machines might reject it, and the nice, sweet volunteers at your county elections office will have to copy your whole ballot onto a clean one, with observers from both major parties watching over her shoulder, so she can run that through the machine and have it counted. If you do need to correct a mistake, the machines can understand if you X out a vote, or if you spill coffee on it, you can go to the elections office and request a replacement ballot.
Other than this, according to the official summary of election laws by the Oregon Secretary of State. “a person may not vote or sign any other person's (including a spouse's) ballot—not even with a power of attorney.”
What you describe does not appear to fall under the one legal exception. My understanding is that, even if Jane is allowed to avail herself of this loophole (for example, because John is unable to read his ballot), officially “Any person assisting a voter must follow the direction of the voter as to how to vote the ballot.” If he only says, “I don’t care,” I suspect she is expected to leave the ballot blank. However, “Just vote for whoever you want,” makes it much more debatable, and I don’t see how it could ever be proven that Jane filled out the secret ballot in some unlawful way.