Here are the top 10 TIMMS rankings for 2010 according to Wikipedia for math, science, and reading.

Programme for International Student Assessment (2009)[1]
(Top 10; OECD members as of the time of the study in boldface)
Maths Sciences Reading
1. People's Republic of China Shanghai, China 600
2.  Singapore 562
3.  Hong Kong, China 555
4.  South Korea 546
5.  Taiwan 543
6.  Finland 541
7.  Liechtenstein 536
8.  Switzerland 534
9.  Japan 529
10.  Canada 527
1. People's Republic of ChinaShanghai, China 575
2.  Finland 554
3.  Hong Kong, China 549
4.  Singapore 542
5.  Japan 539
6.  South Korea 538
7.  New Zealand 532
8.  Canada 529
9.  Estonia 528
10.  Australia 527
1. People's Republic of ChinaShanghai, China 556
2.  South Korea 539
3.  Finland 536
4.  Hong Kong, China 533
5.  Singapore 526
6.  Canada 524
7.  New Zealand 521
8.  Japan 520
9.  Australia 515
10.  Netherlands 508

The countries highlighted in green have some sort of public health care system, either with or with-out a co-pay. Not all of the health care systems are of equal quality, and in New Zealand only hospitals are free (hence the light green shading), not necessarily all of the medical services.

Draw whatever conclusions you like, but at some stage analyzing the data just gets a bit ridiculous.