How do I update a single value in a json document using jq?

Update a value (sets .foo.bar to "new value"):

jq '.foo.bar = "new value"' file.json

Update a value using a variable (sets .foo.bar to "hello"):

variable="hello"; jq --arg variable "$variable" '.foo.bar = $variable' file.json

You set values of an object using the = operator. |= on the other hand is used to update a value. It's a subtle but important difference. The context of the filters changes.

Since you are setting a property to a constant value, use the = operator.

.shipping.local = false

Just note that when setting a value to a property, it doesn't necessarily have to exist. You can add new values easily this way.

.shipping.local = false | .shipping.canada = false | .shipping.mexico = true

a similar function to the operator |= is map. map will be suitable to avoid the requirement of a previous filter for the array...

imagine that your data is an array (very common for this example)

[
  {
    "shipping": {
      "local": true,
      "us": true,
      "us_rate": {
        "amount": "1.00",
        "currency": "USD",
        "symbol": "$"
      }
    }
  },
  {
    "shipping": {
      "local": true,
      "us": true,
      "us_rate": {
        "amount": "1.00",
        "currency": "USD",
        "symbol": "$"
      }
    }
  }
]

hence it is necessary to consider the array in the code as:

http://example.com/shipping.json | jq '.[] | .shipping.local = "new place"' | curl -X PUT http://example.com/shipping.json

or to use the map function that is crafted to work in every array element as

http://example.com/shipping.json | jq 'map(.shipping.local = "new place")' | curl -X PUT http://example.com/shipping.json

Observation

For the sake of those that are learning, you also did some mistakes in the jq usage, just consider that it does "read" the 1st parameter as the program, hence all the desired commands shall be included in the very first string after calling the program.