Stressing and worrying are both several orders of magnitude better than suffering and dying.
Source: My case wasn't a false positive.
So now you have someone who thinks they have might lung cancer, when in fact they probably have almost harmless scars from chicken pox.
But what do you do if you think you have lung cancer? Do you go and get biopsies? The stress from that result is significant and shouldn't be dismissed as something trivial.
My point is people who went through the experience of having test results compatible with cancer and then discover it is just a false positive (which technically it's not even true, there's no cancer diagnosis before a biopsy, it just can't be excluded), well, these very people seem to speak out against screening, or adopting more extensive screening policies, because they had a "false positive", and "it's stressful".
I am pretty sure all the people whose lives were saved by screening find that pretty selfish.
The harm/good ratio of screening depends crucially of the false positive/negative ratio vs. the actual prevalence of the condition or disease being screened for. (Plus, obviously the psychological/medical "cost" of a false negative/positive.) I would recommend the talk at [1]. This particular issue is brought up about midway through the video, but I recommend watching the whole thing.
[1] http://www.ted.com/talks/peter_donnelly_shows_how_stats_fool...
Harms of Screening Mammography: http://www.cancer.gov/cancertopics/pdq/screening/breast/heal...