Which specific statements are false?
I'm glad you asked:
Less will be insured than before
I haven't seen one credible source that claims this. Every credible actuarial report I've seen, and I've seen lots (Milliman, Gorman, Kaiser Family Foundation, etc) indicates that insurance coverage will go up, considerably. The estimates of the increase may have been adjusted down, but that means the increase is smaller, not that the total number is smaller, which this statement claims.
I haven't seen one credible source that claims this. Every credible actuarial report I've seen, and I've seen lots (Milliman, Gorman, Kaiser Family Foundation, etc) indicates that premiums will go, some quite a bit, but average rate increases will be in the double digits and even the most extreme averages will be in the lower triple digits. You might be able to find one outlier with such an incredible increase, but it certainly isn't representative of the population, which your statement would indicate.
insurance will be a lot less affordable than Americans were led to expect,
This is partially true; gross insurance premiums will be much less affordable than initially indicated, but since the demographics the law panders to will get free coverage (Medicaid expansion) or almost free coverage (advance premium tax credits) they will see it as being affordable even though the cost to society isn't. So, really, it's a matter of perspective.
2) fewer people than promised will get insurance and
This is a very fungible claim. There was always a personal choice component in that someone could elect to pay the penalty. If you had said "fewer people than promised will get access to insurance" I would still say it's a fungible claim. Do you consider Medicaid to be "insurance"? Technically, it's not, so the fact that many states won't expand Medicaid means that
more people will get access to insurance since those between 100% and 138% FPL who would have been Medicaid eligible will now be eligible for insurance.
3) millions of people who have coverage through a job now will lose it, thanks to the president’s “reforms.”
This was intentional.
Start with the IRS’s new estimate for what the cheapest family plan will cost by 2016: $20,000 a year to cover two adults and three kids. And that will only cover 60 percent of medical bills, so add hefty out-of-pocket costs, too.
I've already gone over this in another thread, but the article this claim stems from was intellectually dishonest. The IRS never made this claim. The IRS did issue guidelines with examples of how the various calculations worked, and this scenario was an included example, but
that is not the same as actually making this as a factual claim.
The next surprise is for parents who thought their kids would be covered by an employer. Sloppy wording in the law left that unclear until last week, when the IRS ruled that kids won’t be covered.
Starting in 2014, the law will require employers with 50 or more full-time employees to offer coverage or pay a penalty. “Affordable” coverage, that is — meaning the employee can’t be told to contribute more than 9.5 percent of his salary. For example, a worker earning $40,000 a year cannot be required to pay more than $3.800.
But the law doesn’t specifically mandate family coverage — and now the administration says that won’t be required.
The premise of these statements is confused and wrong. The IRS did not rule that kids won't be covered. The administration did not say that family coverage won't be required. ACA §1513(a) amends Chapter 43 of IRC to add §4980H(a)(1), which requires employers to "offer full time employees (and their dependents) the opportunity to enroll in minimum essential coverage". Dependent coverage is clearly afforded under the law.
The quoted statements confuse this matter with the regulations offering definitions on the matter, and then makes incorrect assumptions about the applicability.
In defining "dependents" as used in §4980H(a)(1), the IRS followed existed interpretation to determine that the term included children in most forms but did not include spouses.
The IRS regulations did say that the affordability calculation is based only on the employee.
The combination, then, is that children must be offered coverage, but that coverage need not be "affordable". You could certainly lament this outcome, but I point out that this is clearly not the outcome quoted above to illustrate the incorrectness of the quote.