This is where I stick my neck out and make a bold claim, thinking I am unlikely to be wrong despite very much NOT being an immunologist:

I am not worried and I think they're overreacting, which in the pandemic business is a GOOD thing and in the long run obliterating human contact with explosively susceptible animal reservoirs is a VERY good thing and I am glad they are doing this, but still.

Okay, here we go:

The so called 'mink strains' bear the following mutations in the spike protein:
D614G, just like the whole rest of the world at this point.
Y453F
I692V
M1229I
H69 deletion
V70 deletion

They are interested because this is an unusually large number of mutations to appear in the spike in a short period of time.  Let's look at them

I692V and M1229I are nowhere near the part of the spike where most important antibodies that can neutralize the thing are.  Could change the stability of the trimer or the S1/S2 boundary and thus the behavior/binding, maybe theoretically if they're just right, but I'm calling them as less likely to be immunologically relevant.  They're also not looking like HUGELY disruptive mutations in terms of changes to amno acid properties, maybe.

H69/V70 are in what's called the "n-terminal domain" at the start of the gene.  This domain is up near but not quite at the business end of the spike.  It is not the part that binds ACE2, but is right next door to it.  It is involved in invasion to some degree, and seems to bind to cell-surface sugars a bit while not being the strongest binding thing.  Some other viruses have this domain as their main binding domain but not this one.  About 20% of human neutralizing antibodies are directed against this domain.  Could be slightly immunologically relevant, or could be an adaptation to hit the mink better.

Y453F is actually quite interesting.  This one is in the receptor binding domain and is probably an adaptation to bind the mink or human ACE2 better.  It is also a known 'immune escape' mutant.  DO NOT PANIC at that name.

I will use another, better characterized immune escape mutant to illustrate what this means.  

https://www.biorxiv.org/content/10.1101/2020.11.04.355842v1

There is a mutation circulating in other European strains, N439K.  This one came about as an adaptation to better fit the human ACE2 protein, and has a mildly better binding affinity to the human receptor than the original.  It also is in the receptor binding domain.  Since most neutralizing antibodies attack the receptor binding domain, this renders a subset of antibodies generated against the RBD less effective - in particular, one of the two monoclonal antibodies in the Regeneron cocktail that the president got is rendered completely inert!

Thankfully, and for good evolutionary reason, vertebrate immune systems don't make huge amounts of one or two antibodies against something, they make small amounts of a diverse array of antibodies.  So when an escape mutant of something that's as large a target as this comes along, there's usually other antibodies around that are unaffected by it.  Researchers exposed the serum of hundreds of recovered patients to the original spike and this mutant spike.  They found that most people (~80%) didn't have an appreciable difference in the amount of antibody binding that occurred.  About 7% of the population had the binding ability of their antibodies decline by at least a factor of two, some down by a factor of sixteen - but these were almost exclusively people who mounted a very weak response to begin with, probably because they didn't make many different types.  So a subset of people who already mounted a weak response had their response rendered weaker by this escape mutant.

I should note that all the vaccines in phase three trials are generating immune reactions at least as strong as the strongest convalescent patients did, and some are up in the 2-3x as strong range.

Escape mutants will come along, but the fact that it's been a year and we are only just starting to see weak escape mutants tells me that this is gonna happen WAY slower than flu.

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So, they saw a spike protein in this secondary zoonotic strain with many mutations, and there is word that they tested it against monoclonal antibodies and it escapes a number of them.  Word floating around on virology twitter pre-publication is that upon exposure to actual patient serum, it again weakens response for some but is still very well neutralized by most.  I suspect they are being paranoid about a spike with multiple mutations being out there as as base for further evolution, and don't want more events like this happening in the future in this highly susceptible animal reservoir.  Could be that that deletion in the N-terminal domain is immunologically relevant even though that domain is less relevant than the receptor binding domain most of the time and thus that there's a double-escape floating around now.

I VERY much look forward to the actual publication of actual results.  And am prepared to eat my words.

EDIT:  The data is finally available!

https://files.ssi.dk/Mink-cluster-5-short-report_AFO2

We were warned that this SARS-related virus was likely to mutate and change much like seasonal flu.  Consequently, to expect a silver bullet in the form of a vaccine is likely unrealistic.