Don't know if the numbers would be high enough to move the employment needle all that much. For example, Switzerland, one of the earliest adopters, had 750 assisted suicides in 2014 out of 8M inhabitants.
(I'll use A.S. as an abbreviation below)
In fact, reading from the wikipedia link, you could even generate extra employment in the suicide industry:
Euthanasia organisations have been widely used by foreigners, in what critics have termed suicide tourism. As of 2008, 60% of the total number of suicides assisted by the organisation Dignitas had been Germans.
However, looking at Canada's official stats, the numbers are much higher (only residents can apply):
In 2020, there were 7,595 cases of MAID reported in Canada, accounting for 2.5% of all deaths in Canada.
The number of cases of MAID in 2020 represents a growth rate of 34.2% over 2019. All provinces experienced a steady year over year growth during 2020.
When all data sources are considered, the total number of medically assisted deaths reported in Canada since the enactment of federal legislation in mid-2016 is 21,589.
Still not sure that's really that significant on a population of 36M. And a lot of the facilities taking on these clients are probably understaffed anyway, without being able to attract more staff, so again not sure if there'd be a big drop.
The Netherlands have similar, low, but also rising numbers:
Official data show that the number of euthanasia cases has risen more or less continuously since 2006, reaching 6361 in 2019. These cases make up just a small proportion of all deaths, but they have doubled from just under 2% in 2002 to just over 4% in 2019.
Also, you have a coordination/communication problem if that was really the reason: if entity X wants to limit assisted suicide due to lower employment in associated industries, then does it want to go on the record doing so: "Can't have assisted suicide because it would lower employment"??? This is already a contentious issue between the free will/personal choice people vs. the sanctity of life/religious ones. Making it public that one goal is to "support employment" would look very very mercenary. So, assuming a union did want to stop A.S., it would probably argue on any other ground than this and keep their true motivations to themselves.
Here's a snippet from BBC in the context of a shift from opposition to neutrality.:
Of palliative medicine doctors, 80% voted to maintain the college's opposition to assisted dying.
No details given as to the reason.
Another link to the UK, more critical, but again no reason given.
Second, the core justification for the ‘blanket ban’ has deflected attention away from the interests of the medical profession itself, which in the United Kingdom has a long tradition of organized and powerful opposition to assisted suicide.
Here's a pro-A.S. paper, which objects to some physicians' opposition, but does not claim it happens on those grounds.
Or, in India, lots of anti A.S. arguments, none crassly commercial is stated.
AndAgain, another UK position: nothing $$$ oriented.
Finally, that is also not to say health care unions can't have other valid concerns about the responsibilities their members would be expected to take on if A.S. gets adopted. And that might include wishes to be able to opt out individually.