I think the false negative rate is wrong in that post. The original source says

The BCSC data indicate that false-positive mammography results are common in all age groups. The rate is highest among women age 40-49 years (97.8 per 1,000 women per screening round) and declines with each subsequent age decade (Table 7). The rate of false-negative mammography results is lowest among women age 40-49 years (1.0 per 1,000 women per screening round) and increases slightly with subsequent age decades.

Which suggests to me that P(negative|cancer) is not 1/1000 but 1/(actual cancer rate per thousand) which appears to be around 1/4 from the numbers in the paper. The false negative rate given here of 'up to 20%' seems much more in line with that interpretation than does the 1/1000 false negative rate.

The wording of the original report is quite misleading as it suggests the false negative rate increases with age but I think they actually mean that the number of false negatives per 1000 increases (because the cancer rate is increasing). The other link suggests that P(negative|cancer) is higher with younger women due to firmer breast tissue making it harder to distinguish a tumor from healthy tissue. Other pages I found through Google suggested the same.