You are wiser than most to realize that there is a distinction between Constitutionally granted rights and what many people are now declaring is a right.
You are right in a sense, but I think you are seeing "rights" too narrowly. The distinction that is generally made is between classical, negative rights (the ones in your 'bill of rights') and the positive, modern, or social rights (such as in your "equal protection" clause and in many european constitutions and also in some articles of the universal and European declarations of human rights.
The right to free speech is a "classic" (human) right. It is something between you and the government, and it acts as a prohibition on the government. It is a 'negative right' in that the government isn't forced to *do* anything, they're forced to not do stuff, like putting you in prison if you say stuff. The bill of rights used the phrase "congress shall pass no law" for a reason. The right doesn't mean that the government has to pay publishing cost for your newspaper, and it also doesn't mean that other citizens are forced to listen to you or banned from telling you that you're a twit if you 'exercise' your basic right. These rights can be "absolute" in the sense that they allow no exceptions, although it often isn't (the classical example being yelling 'fire!' in a crowded cinema) and in principle they cannot clash, since every negative right bans the government from doing something. The right to the freedom of the press and the freedom of religion do not clash: if you exercise your press freedom to say that not all Catholics are holy, there is no clash of rights, since the freedom of religion never prohibited you from saying that, it simply prohibits the government from establishing, favoring, regulating, or banning any religion.
The right to healthcare (and education, housing, property non-discrimination, etc) are all "social" (human) rights. They are generally positive rights, where the government has to provide something for the citizen. These rights are never absolute, in the sense that it is obvious that no one can receive all the education, health care, etc that money can buy. Also, these rights can clash with each other and with the classical rights: my right to self-expression can clash with your right to non-discrimination.
Sometimes, a right can be both 'positive' and 'negative'. For example, the right to "life" in the universal and European declaration of human rights (and implied in your declaration of independence) is mainly negative, in prohibiting the government of killing you in most circumstances, but can (and is) also interpreted positively as a positive duty to prevent certain loss of life and/or to investigate suspicious death. Similarly, the right to own property (e.g. article 17 UDHR but see also the 3d-5th amendment in the US Bill of Rights) is a negative right in prohibiting the government from taking your stuff (except through taxes, eminent domain etc.). However, it also implies a duty for the government to protect your property by banning and investigating theft etc, although the US Bill of Rights is phrased to exclude those duties by listing prohibitions against search, seizure, and quartering by the government, and does no list a "right to property".