During sports labor negotiations, there is no group that's as left out of things as the fans. It's during these times that the fans realize exactly what they are to a business - nothing but walking wallets. The entire sports industry is built upon the backs of the fans, but they have no say in anything that goes on.
The players are who the fans come and see, and they don't get much respect from the owners as a group. But at least they have a voice in what goes on. Both the owners and the players' union would be nothing without the fans, and yet they're dismissed and completely ignored from participating in these labor discussions.
However, it may be to the players' union's benefit to actively encourage the fans to support them.
What it all comes down to is money. The players have the most to lose in this contest of wills, despite being the more accommodating of the two sides. However, if they encourage the fans to stop buying jerseys and other merchandise, cancel season tickets, and not go to opening night in protest when the season finally does start, then they have an ally that may tip the scales in their favor.
Of course, in the short run, this will hurt the players. Fans not going to games or buying merchandise just cuts into their team's business and may affect future individual salary negotiations. But in the long run, it will give them the leverage they need to prevent future lockouts.
The bottom line is that, if the owners know that fans won't show up, then they'll be more loathe to threaten with a lockout when they don't get their way. But because they know that the fans will show up, regardless of what may happen, they're perfectly okay with having a lockout to try to get what they want. The key to the whole thing are the fans, if anyone can get them all to go in the same direction at once.
The majority of fans are already on the players' union side in the current situation, so encouraging them to not spend money on hockey will just give fans a legitimate and organized way in which to show their anger over the situation.
Getting all NHL fans to do something together all on their own would be like herding cats. You might be able to get a single fan base to do it, but you can't get all of them all to do the same thing at the same time if it comes from within. If the NHLPA suggests and encourages it, however, likely more would join in than any grassroots effort could probably achieve.
What it comes down to is when you're fighting over who gets the bigger slice of the pie, usually the side that's friends with the baker wins.