You can always ask for more.
You may or may not get it.
Negotiation is always a game of incomplete information on both sides.
You don't know:
- Where the offer lies with respect to the compensation of current employees
- How this insurance plan rates in the history of the company (for all you know, the insurance planning folks just made a lame insurance deal and the salary folks don't even know how lame it is for the industry)
- How this offer stacks up compared to the person they hired into this role last week
- How hard or easy it has been to find candidates
They don't know:
- How badly the insurance deal impacts your particular case (nor should they - some of this stuff is medical and they have no right to ask)
- How much this improves/harms your overall financial state compared to your current situation
- What other offers you may have received
Either side may expose more information for the negotiation, neither side is obligated to. You'll do better in the negotiation the more clear you can be and the more flexible you can be - but that only goes as far as is useful to you.
Mention the insurance shortage if having BETTER insurance would be as (or more) beneficial to you as more salary. If you really don't care about the insurance, you'd rather have the money - then speak in generalities of "total compensation".
Mention other offers of the table if they are actually in hand or 99% likely. Realize that the company may draw a line and say "no" - just because someone else will pay you more doesn't mean this company will pay you more.
You can certainly reiterate why you think you'd be a great asset to the company, but in all honesty, if the company has made the offer, the people who know why a given fact makes you a great asset may be highly separated from the negotiation process. For example, as a hiring manager I frequently say to my recruiter "see if you can get this guy on board for a range between X and Y" - my recruiter tries. If he nails the deal, great. If not he gets a new ball park and we debate - generally the bigger the difference, the more management I have to get on board with why this person is worth it. At that point, if the person is telling the recruiter "did I mention I also know ABC technology and have experience with DEF business skill?" - the information will be lost in transmission as I am trying to say to my management "seriously, Y rate + this much more is what we need to get the right person". And if it's small money then we just close the deal and move on.
What I DO expect is that my recruiter may come to me with creative ways to make a better package - for example, "You said to hire no higher than $91K, this guy wants $93 but it's because he says our insurance is lame. It turns out, we just moved into an HSA program for all new hires, and that means that folks have a much bigger insurance load for families than they used to - I see that current employees will be grandfathered into our old program - I checked with the insurance folks and they found this loop hole - can you do the manager paperwork that makes the loop hole viable for this hire?" - to which my answer is "YES! where do I sign?" or "meh, sounds like hard work, give the guy an extra $2K".
A lot of this, is both corporate structures and also the range of salary difference. It's a fair point that in cases of total compensation, the difference in packages can seriously add up to $10K+ of lost value. While that's true, the negotiation and approval process at $2K is a lot different than $10K+ - so a single insurance package is unlikely to fix the problem... instead if the candidate has a reason, the reason has to be something more like "I have an offer from your competition" or "Did I mention I am also a secret assassin able to solve problems 20 times faster than normal people? ... I didn't put that on my resume, because I didn't realize until recently that you were qualified for receiving the benefits of secret assassins"