Learn English – Is this “that” a conjunction, or something else

Certain nouns in English can take declarative content clauses as Complements.

• the idea [that interplanetary travel might be possible]
• the notion [that all people are created equal]
• the argument [that the chicken came first]
• any evidence [that he had eaten the elephant]

The bracketed strings in the examples above are all content clauses (as opposed to relative clauses or comparative clauses). The word that here is a subordinator (read 'subordinating conjunction' or 'complementizer'). It is a marker indicating that the following finite clause is a subordinate clause.

The noun statement is one of these nouns which can take a content clause as Complement.

• All papers should be accompanied by a statement [that they have not already been published elsewhere]

However, when the noun statement is used in this way, it is referring to an illocutionary act. In other words the noun statement refers to the physical act of stating (whether by being put in writing or by speaking).

When the noun statement refers to a physical piece of paper, for example, rather than the actual declaring of the statement itself, it is much rarer for the noun statement to take a content clause as Complement:

• He signed a statement that he hadn't done it.

Many readers might find the sentence above a bit clunky, myself included. In such an instance, they might find it preferable to use a gerund-participle clause to modify the noun:

• He signed a statement saying that he hadn't done it.

If the noun statement is the subject of the sentence then this contrast between the acceptability of a content clause depending on the exact meaning of the word statement becomes clearer:

• His statement that he was going to ratify the treaty was well received.
• ?The statement that he was going to ratify the treaty was photocopied.
• The statement saying that he was going to ratify the treaty was photocopied.

The first example above in which statement refers to the communicative act works well with a content clause. However, the second, in which statement is best thought of as some kind of document, does not work so well. It is awkward. The third example which uses a clause headed by the verb saying is an improvement.

The Original Poster's example

We will enroll you unless you sign a statement that you do not want us to.

This sentence is marginally acceptable. However, because of the semi-physical meaning of the word statement here, I concur with the Original Poster that the sentence would be much better if it used a gerund-participle clause:

We will enroll you unless you sign a statement saying that you do not want us to.